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%I #12 Feb 15 2022 20:59:18
%S 1,1,2,1,3,1,4,1,2,2,5,0,6,3,5,1,7,0,8,1,7,4,9,-1,3,5,2,2,10,0,11,1,9,
%T 6,11,-2,12,7,11,0,13,1,14,3,4,8,15,-2,4,1,13,4,16,-1,14,1,15,9,17,-5,
%U 18,10,6,1,17,2,19,5,17,4,20,-4,21,11,4,6,19,3
%N Let f be multiplicative with f(prime(k)^e) = k + e*i for any k, e > 0 (where i denotes the imaginary unit); a(n) is the real part of f(n). See A351465 for the imaginary part.
%C Apparently, each integer (from Z) appears in this sequence.
%e For n = 42:
%e - 42 = 2 * 3 * 7 = prime(1)^1 * prime(2)^1 * prime(4)^1,
%e - f(42) = (1+i) * (2+i) * (4+i) = 1 + 13*i,
%e - and a(42) = 1.
%p b:= proc(n) option remember; uses numtheory;
%p mul(pi(i[1])+i[2]*I, i=ifactors(n)[2])
%p end:
%p a:= n-> Re(b(n)):
%p seq(a(n), n=1..78); # _Alois P. Heinz_, Feb 15 2022
%t f[p_, e_] := PrimePi[p] + e*I; a[1] = 1; a[n_] := Re[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* _Amiram Eldar_, Feb 15 2022 *)
%o (PARI) a(n) = { my (f=factor(n), p=f[,1]~, e=f[,2]~); real(prod (k=1, #p, primepi(p[k]) + I*e[k])) }
%Y Cf. A289310, A351465, A351475.
%K sign,easy
%O 1,3
%A _Rémy Sigrist_, Feb 11 2022
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